What numbers were used to create a data set with six numbers that have a mean of 40 a median of 41 and a range of 18?

Answer 1

There are an infinite number of possible answers. One answer is {32, 34, 40, 42, 42, 50}

The reason there are an infinite number of answers is that you basically have six variables, a, b, c, d, e, f. Assuming these are in numerical order, the assumptions translate to... (a + b + c + d + e + f) / 6 = 240

(c + d) / 2 = 41

f - a = 18

That's six unknowns with only three equations, so it's under-determined and has infinitely many solutions in real numbers. For example, in addition to the above solution, you have

{32.1, 33.9, 40, 42, 41.9, 50.1} and you can keep tweaking in that manner indefinitely.

However, if you restrict the number to be positive integers (that is, >=1), then there are only a finite number of solutions. and in fact, very few. This is because the range constraint confines the solution to a relatively small range of values, and the median and average constraints assure that those values must cluster around 40-41.

If you require that the numbers are all different, there are even fewer solutions.